g] consisting of one vertex v with g self-loops, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (where [alpha] = [alpha] [member of] N) is (conjecturally) the dimension of the middle cohomology group
of a character variety parameterizing certain representations of the fundamental group of a closed genus-g Riemann surface to [GL.
Knowing that the coherent cohomology of polydisks vanishes also opens the road towards computing global cohomology groups
for projective analytic spaces over ring of integers of number fields.
In , we studied Lie-Rinehart cohomology of singularities, and we gave an interpretation of these cohomology groups
in terms of integrable connections on modules of rank one defined on the given singularities.
The orbicycle index polynomial can be use to compute the even dimensions of the orbifold cohomology groups
for global orbifolds of the form [M.
It surveys several algebraic invariants: the fundamental group, singular and Cech homology groups, and a variety of cohomology groups
Their topics include moduli spaces of twisted sheaves on a projective variety, integral Hodge classes on uni-ruled or Calabi-Yau threefolds, birational geometry of symplectic resolutions of nilpotent orbits, moduli stacks of second-rank Giesker bundles with a fixed determinate on a nodal curve, vector bundles on curves and theta functions, Abelian varieties with bounded modular height, the moduli of regular holonomic Dx-modules with natural parabolic stability, cohomology groups
of stable quasi-Abelian degenerations, semi-stable extensions on arithmetic surfaces, cusp form motives, polarized K3 surfaces, rigid geometry and applications, and moduli of stable parabolic conventions with Riemann-Hilbert correspondence and other features.